The Penalized Analytic Center Estimator

• Published in: Econometric reviews Vol. 35; no. 8-10; pp. 1471 - 1484
• Main Author: Knight, Keith
• Format: Journal Article
• Language: English
• Published: New York Taylor & Francis 25.11.2016; Taylor & Francis Ltd
• Subjects: Analytic center; C210; Dantzig selector; Econometrics; Economic models; Estimators; Lasso; Mathematical analysis; Mathematical models; Norms; Regression; Regression analysis; Regression coefficients; Selectors; Shrinkage estimation; Tuning
• ISSN: 0747-4938; 1532-4168
• DOI: 10.1080/07474938.2015.1092800

In a linear regression model, the Dantzig selector (Candès and Tao, 2007 ) minimizes the L 1 norm of the regression coefficients subject to a bound λ on the L ∞ norm of the covariances between the predictors and the residuals; the resulting estimator is the solution of a linear program, which may be nonunique or unstable. We propose a regularized alternative to the Dantzig selector. These estimators (which depend on λ and an additional tuning parameter r) minimize objective functions that are the sum of the L 1 norm of the regression coefficients plus r times the logarithmic potential function of the Dantzig selector constraints, and can be viewed as penalized analytic centers of the latter constraints. The tuning parameter r controls the smoothness of the estimators as functions of λ and, when λ is sufficiently large, the estimators depend approximately on r and λ via r/λ 2 .